Separation of variables for quantum harmonic oscillator

[jaejoon89]

a) Show that the Hamiltonian for the quantum harmonic oscillator in 3D is separable, b) calculate the energy levels.----a) If it's separable H = H_x + H_y + H_z, so do I just re-arrange the kinetic and potential terms of the Hamiltonian in this case? that seems kind of trivial, as if I'm probably missing something...

b) I assume that if the Hamiltonian is separable, we can use the method of separation of variables to find the energy, i.e., Psi(x,y,z) = X(x)Y(y)Z(z), substitute and then divide by it to find the three energies. So I get, for example,

(1/(X(x))*(-hbar^2 / 2m d^2 / dx^2 + 1/2 k_x x^2)X

How does that correspond to the energy level of the oscillator for the x coordinate, i.e.

hbar*omega (n + 1/2) where n is an integer

 

[Jhero]

?a) Yes, you are correct. The Hamiltonian for the quantum harmonic oscillator in 3D can be written as H = H_x + H_y + H_z, where H_x, H_y, and H_z are the Hamiltonians for the oscillator in the x, y, and z directions respectively. This is because the potential energy for the harmonic oscillator is separable in the three spatial dimensions, meaning it can be written as a sum of terms that only depend on one coordinate. Therefore, the Hamiltonian is also separable.

b) To calculate the energy levels, we can use the method of separation of variables as you mentioned. This involves writing the wavefunction as a product of three separate functions, one for each coordinate, and then plugging this into the Schrodinger equation. This will result in three separate equations, each of which can be solved for the energy eigenvalues.

For example, for the x coordinate, we have:

(-hbar^2 / 2m d^2 / dx^2 + 1/2 k_x x^2)X = EX

where E is the energy eigenvalue for the x coordinate. This is a simple differential equation that can be solved by using the substitution u = x^2. The solution gives us the energy eigenvalues for the x coordinate as:

E_x,n = (n + 1/2)hbar*omega_x

where n is an integer and omega_x is the frequency of the oscillator in the x direction.

Similarly, we can solve for the energy eigenvalues for the y and z coordinates, and then combine them to get the total energy eigenvalue for the 3D harmonic oscillator:

E_n = E_x,n + E_y,n + E_z,n = (n + 3/2)hbar*omega

where omega is the frequency of the oscillator in all three directions. Notice that this is the same expression as the energy level of the oscillator for a single coordinate, but with an additional 1/2 factor due to the three-dimensional nature of the problem.

In summary, by using the method of separation of variables, we can show that the Hamiltonian for the quantum harmonic oscillator in 3D is separable and calculate the energy levels for the oscillator in all three coordinates.